TRANDES: A FORTRAN PROGRAM ,FOR TRANSONIC AIRFOIL ANALYSIS OR DESIGN q

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TRANDES: A FORTRAN PROGRAM
,FOR TRANSONIC AIRFOIL
ANALYSIS OR DESIGN
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Prepared by
TEXAS A&M UNIVERSITY
College Statlon, Tex. 77843
for Langley Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. JUNE 1977
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1. Report No.- 2. Government Accession No. 3. Recipient’s Catalog No.
NASA CR-2821
4. Title and Subtitle 5. Report Date
June 1977
TRANDES: A Fortran Program for Transonic Airfoil Analysis
or Design 6. Performing Organization Code
7. Author(s) 6. Performing Orqamzation Report No.
Leland A. Carlson
10. Work Unit No.
9. Performing Organization Name and Address 505-06-33-08-00
Texas A&M University
11. Contract or Grant No.
College Station, Texas 77843 NSG-1174
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address Contractor Report
National Aeronautics and Space Administration
14. Sponsoring Agency Code
Washington, DC 20546
15. Supplementary Notes
Final report
Langley Technical Monitor: Jerry C. South, Jr.
16. Abstract
A program called TRANDES is presented that can be used for the analysis of steady, irrotational,
transonic flow over specified two-dimensional airfoils in free air or for the design of airfoils
having a prescribed pressure distribution, including the effects of weak viscous interaction
(i.e., massive separation is not considered). Instructions on program usage, listings of the
program, and sample cases are given.
-.
17. Key Words (Suggerted by Authoris)) 18. Distribution Statement
Supercritical airfoil design Unclassified - Unlimited
Viscous interaction
Relaxation
Cartesian coordinates
Subject Category 02
IQ. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Rice’
Unclassified Unclassified 109 $5.50
*For sale by the Netionel Technical Information Service, Springfield, Virginie 22161
TRANDES: A FORTRAN PROGRAMF OR TRANSONIC
AIRFOIL ANALYSIS OR DESIGN
BY
Leland A. Carlson
Texas A&M University
SUMMARY
A program called TRANDES is presented that can be used for the analysis of
steady, irrotational, transonic flow over specified two-dimensional airfoils in
free air or for the design of airfoils having a prescribed pressure distribu-
tion, including the effects of weak (no massive separation) viscous interaction.
Instructions on program usage, listings of the program, and sample cases are
given.
INTRODUCTION
The program described in this report can be used for either the analysis of
the flowfield about an airfoil in free air or for the design of an airfoil
having a specified pressure distribution. In the direct or analysis mode the
airfoil shape is prescribed and the flowfield and surface pressures are deter-
mined. In the direct-inverse or design mode an initial nose shape is given
along with the pressure distribution on the remainder of the airfoil, and the
flowfield and actual airfoil shape are computed. In either case, the effects of
weak viscous interaction may be included at the option of the user.
The program solves the exact equation for the perturbation potential in a
Cartesian coordinate system. Most of the,background about the equations solved,
formulation of the boundary conditions , and the difference scheme used is given
in references 1-3. This report gives instructions on the use of the computer
program and also some additional details concerning the inclusion of weak vis-
cous interaction. It should be noted that in this report the term weak viscous
interaction implies that there is no massive boundary layer separation on the
airfoil. Nevertheless, for aft-loaded airfoils at transonic speeds, the effect
of viscous interaction on airfoil performance may still be quite large.
The next section gives a general description of the problem and the method
of solution. Then the instructions for using the computer program,and a de-
scription of the inputs and outputs are given. The appendices contain
additional details as well as listings of the program and the sample cases.
GENERAL DESCRIPTION
The program described in this paper obtains the inviscid flowfield by
solving the full , inviscid, perturbation-potential flow equation in a Cartesian
grid system. This system which is usually aligned relative to the airfoil
chord line, has been found to efficiently yield accurate solutions for biconvex,
conventional, and aft-cambered airfoils. In the program, a stretching is ap-
plied to the coordinates such that the infinite physical plane is mapped to a
finite computational space. Thus, the boundary conditions at infinity can be
applied directly and there is no need for an asymptotic far-field solution.
Details about the stretching functions are given in appendix A.
The method of solution is to replace the governing second-order partial
differential equation with a non-conservative system of finite difference eq-
uations that includes,at supersonic points, a form of Jameson's "rotated" dif-
ference scheme (ref.4). The difference equations are then solved by column
relaxation, which in order to obtain rapid convergence is usually-done on sev-
eral different grids. In the analysis case, the difference equations are first
solved on a very coarse grid (typically 13x7). The solution is then interpo-
lated and used as an initial condition for a coarse grid (typically 25x13).
This procedure can be repeated twice more to obtain solutions on a medium grid
(49x25) and on a fine grid (97X49). The latter has 130 points on the airfoil;
however, excellent results are usually obtained on the medium grid, particularly
considering the computer time involved. For typical examples see reference 1.
In the inverse case, which is normally used for airfoil modification or
design, an initial airfoil shape must be assumed. However, this choice is not
critical, and the final airfoil shape may be considerably different. Since exi
perience indicates that the inverse scheme works best if the perturbation po-
tentials have reasonable initial values, fifty relaxation cycles are first per-
formed in the direct mode for the initial airfoil shape on a very coarse grid.
The grid is then halved and the inverse procedure initiated using the input
pressure distribution as the boundary condition in the inverse region. As in
the direct case, the grid may be refined again to the medium grid (typically
49x25) where the results are usually adequate. Fine grid usage in the inverse
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case is not recommended due to slow convergence. On each grid the airfoil
shape is recomputed every ten relaxation cycles after the first fifty.
After the solution has been obtained in the design case, the resultant
shpae is treated as a displacement surface and the displacement thickness is
automatically subtracted to obtain the actual airfoil surface. The boundary
layer charactqristics are determined by the Nash-Macdonald method (ref.5)
with smoothing.
The effects of viscous interaction may also be included in analysis cases
at the option of the user. To preserve numerical consistency with the inverse
scheme, the Nash-Macdonald method is also used in such 'analysis cases, starting
with the 50th cycle on the coarse grid. At that point the displacement thick-
ness is computed at the same x coordinates as the inviscid grid and the dis-
placement surface ordinates updated using under-relaxation. The slopes are
then determined from cubic splines through the new ordinates, which are updated
by a new boundary layer calculation every ten relaxation cycles thereafter.
For those cases having extensive trailing edge separation an empirical boundary
layer correction is available. However, it is not necessary for most cases.
It should be noted that while the program can include the effects of bound-
ary layer interaction, no correction has been applied for the effects of wake
curvature and an empirical approach has been used in the trailing edge region.
Thus, the results should be viewed as pressure versus lift coefficient, moment
vs. lift, etc. instead of angle of attack. However, the error in angle of
attack is believed to be small.
Typical total computation times on an Amdahl 47O/V6 or a CYBER 175-T are
60-70 seconds for medium grid results and less than 250 seconds for fine grid
solutions.
PROGRAMU SAGE
The program is written in FORTRAN IV programming language for use on IBM
360-370, Amdahl 470, CDC 6600, and CDC CYBER series computers. The program can
be overlaid in order to reduce computer storage, if required. In nonoverlay
mode it requires less than 200,000 bytes on an IBM type machine. Some modifi-
cation to formats etc. may be required to run the program on different computer
systems.
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The input cards are summarized in the following table:
Read
Variables Format
Order
1 NTITLE 20A4
2 NAMELIST/FINP/M,W,Xl,X2,ALP,EPS,EPSS, Namelist
X4,S4,CONV,Al ,AP,A3,RN,XIBDLY ,CIR,CDCORR,
RDEL,RDELFN,SP,XSEP,XLSEP,XPC
3 NAMELIST/IINP/IMAX,JMAX,IKASE,INV,MITER, Namelist
NHALF,ITACT,ISKP2,ISKP3,ISKP4,ITERP,IREAD,
LP,ITEUPC,ITELWC
4 P(I,J) I=l,IMAX; J=l ,JMAX (Only if IREAD=l) 5E15.7
5 PB(1) I=1 ,IMAX (Only if IREAD=l) 5E15.7
6* Xl ,x2 2F10.5
7 NI 15
8 XI(I),YI(I), I=l,NI 8F10.4
9 DERIX,DERIY ,DERFX,DERFY 8F10.4
10 NIB 15
11 XIB(I),YIB(I), I=l, NIB 8F10.4
12 DERIXB,DERIYB,DERFXB,DERFYB 8F10.4
13* Xl ,x2 2F10.5
14* CPU(I), I=11 , ITE 8F10.3
15* CPL(I), I=Il, ITE 8F10.3
16* x1,x2 2F10.5
17* CPU(I), I=11 ,ITE 8F10.3
18* CPL(I), I=Il, ITE 8F10.3
19* Xl ,x2 2F10.5
20* CPU(I), I=Il, ITE 8F10.3
4
21* CPL(I), I=Il, ITE 8F10.3
* Read only in the design mode when INV=l
NOTE: In the design mode steps 13-15 are for the coarse grid, 16-18
for the medium and 19-21 for the fine grid (if used).
The definitions of these input variables are as follows:
NTITLE - Description of case. Up to 80 alphanumeric characters. *Appears on
printed output, at the beginning of the results for each grid.
M - Freestream Mach number (real variable). Default 0.5
W - Relaxation factor for subsonic points. Should be in the range O<W*2.0
Default 1.7
Xl - X location where direct calculation stops. In analysis mode it should
be set to 0.5 (i.e. trailing edge). In design mode it is usually set
to slightly less than the third point from the leading edge or larger.
Default 0.5
x2 - End of the inverse region. For analysis case set to a large number.
In inverse design case set to 0.5 (i.e. trailing edge). Default
10000.0
ALP - Angle of attack in degrees. Default 0.0
EPS - Subsonic damping factor to match difference equations at sonic line if
needed. EPS has no effect on accuracy of solution, only on stability
and convergence rate. Normally it is not needed. Defau 1t 0.0
EPSS - Supersonic damping factor for iterative stability. Note that EPSS has
no effect on the accuracy of the converged solution, only on the sta-
bility and convergence rate..EPSS should typically be about M2max-1,
where Mm,, is the maximum local Mach number. Default 0.4
x4 - The positive X location where the coordinate stretching changes. It
should be near the airfoil trailing edge. Default 0.49.
s4 - The positive 5 value in the computational plane where the stretching
changes. Default 2.0
CONV - Convergence criteria control value. Iterations stop when the maximum
change in the perturbation potential (between relaxation cycles) is
less than CONV. Default l.E-05
Al - Stretching constant for the Y direction. It can be used to control AY
and An near the horizontal axis. It is usually best to have A5 = AQ
5
near the leading edge of the airfoil. Default 0.246
A2 - First stretching constant for the X-direction. It is equivalent to
f (g) at 5= c4. The value of A2 determines the horizontal step size
near the leading and trailing edges, i.e.
See Appendix A. Default 0.15
A3 - Second stretching constant for the x-direction. It determines the
physical location of the vertical grid line adjacent to grid side
edge. Default 3.87.
RN - Freestream Reynolds number based on chord length. Used only when
viscous interaction included. Default 2O.E+06.
XIBDLY - The x-location at which transition is assumed to occur. The turbulent
boundary layer calculation starts at the next grid point. The rela-
tionship to percent chord is
XIBDLY = (X chord-50.0)/100.0
Default -0.44.
CIR - Circulation about airfoil. If an initial solution is inputted, it
must be the corresponding value of circulation. (CIR = CL/2.0).
Default 0.0
CDCORR - Correction to the computed wave drag coefficient for the finest grid
used. Because of the lack of a large number of points in the leading
and trailing edge regions, the wave drag coefficient has an error as-
sociated with grid size, spacing, and lift coefficient. The magnitude
of CDCORRa s a function of lift can be determined from a series of
calculations at different angles of attack at subcritical speeds, where
the wave drag should be zero. Note that the correction should be de-
termined for each airfoil and grid combination. Default 0.0. See Ap-
pendix B.
RDEL - Relaxation parameter for the boundary layer displacement thickness.
It is used only when viscous interaction is included and IMAX 2 55.
Default 0. 25
RDELFN - Fine grid relaxation parameter for the boundary layer displacement
thickness. It is used only when viscous interaction is included and
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IMAX > 55. Default 0.125
SP - Maximum value allowed for the Nash-Macdonald separation parameter when
x < XSEP. Used only in the viscous interaction case. Default 0.004.
XSEP - X location after which the Nash-Macdonald separation parameter can as-
sume its calculated value. Used only in the viscous interaction case.
Default 0.44
XLSEP - Location at which the trailing edge correction procedure begins. It
should correspond to the point of separation, if used. Between XLSEP
and the trailing edge the pressure distribution and the displacement
surface is modified. Used only if ITEUPC and/or ITELWC equal 1.
Default 0.50
XPC - Location after which the'lower surface displacement thickness is re-
quired to continue decreasing once it has started to decrease. Up-
stream of XPCthe displacement thickness is required to be monoton-
ically increasing. For most aft-cambered airfoils it should be 0.1
and in conventional airfoils it should be 0.5. Default 0.1
IMAX - Number of vertical grid lines in the horizontal direction. I = 1 is
upstream infinity and I = IMAX is downstream infinity. For each grid
refinement IMAX is increased such that IMAXnew = 2(IMAXold) -.l.
The limit on IMAX is 99. Default for use on first grid is 13.
JMAX - Number of horizontal grid lines in the vertical direction. J = 1
corresponds to infinity below the airfoil- and J = JMAX is infinity
above the airfoil. The same formula and limit that apply to IMAX also
apply to JMAX. Default 7.
IKASE - An integer number describing the case. It is limited to a maximum of
six digits. Default 100.
INV - Parameter determining program mode. It should be zero for analysis
cases and one for inverse design cases. Default 0.
MITER - Maximum number of iterations (complete relaxation cycles) allowed on
first grid. MITER is halved for each grid refinement. However, on
the fourth grid, MITER is reset to 400. Default 800:
NHALF - Number of grid refinements to be done. Default 2.
ITACT - Viscous interaction control parameter. It should be set to zero for
analysis cases without interaction and for design cases. It should be
one for analysis cases with interaction. Default d.
7
ISKP2 - Airfoil update control parameter for grid two. It should be O.if on
grid two an update is desired every 10 iterations. It should be 1 if
an update is not desired until the grid two solution is completed.
Only used in the inverse design mode. Default 0.
ISKP3 - Same as ISKP2 but for grid 3 (medium grid).
ISKP4 - Same as ISKP2 but for grid 4 (fine grid).
ITERP - Interpolation parameter. If in the design mode the input Cp distribu-
tion for the grid 4 is to be read in, ITERP should be 0. If it is
desired to linearly interpolate the Cp distribution of grid 3, it
should be 1. Default 0.
I READ - Starting solution control parameter. If IREAD is 0, the initial per-
turbation solution is assumed to everywhere be zero. If it is 1, an
initial solution is read in from data cards. Default 0.
LP - Relaxation cycle interval at which boundary layer, surface ordinates,
etc. details are printed. Useful for diagnostics. Default 1000.
(No printout.)
ITEUPC - Upper surface'trailing edge correction control parameter. If trailing
edge correction desired, ITEUPC should be 1. If not it should be zero.
Only used in the viscous interaction case. Normally the correction
is not needed. Default 0.
ITELWC - Lower surface trailing edge correction control parameter. If correc;
tion desired, ITELWC should be 1. If not it should be 0. Only used
in the viscous interaction case, and normally,the correction is not
needed. Default 0.
P(I,J) - Nondimensional perturbation potential, 4ij, at point 1,J.
PB(1) - Nondimensional perturbation potential at point I on the y=O'grid line.
Xl, x2 - Same definition as above. However, in the inverse design case they
must be read in prior to the solution of each grid. On the first grid
(step 6 in above table) should use X1=0.5, X2=10000.0. On remaining
grids (steps 13,16, and 19), Xl should be the location where the
direct calculation stops and X2 should be 0.5.
NI - The number of coordinate pairs used to describe the upper surface of
the airfoil. Presently limited to 99.
XI(I) - Input coordinates in the horizontal direction for the airfoil upper
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